Classifying space for O(n)

In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space $\mathbb {R} ^{\infty }$.

Cohomology ring

The cohomology ring of $\operatorname {BO} (n)$ with coefficients in the field $\mathbb {Z} _{2}$ of two elements is generated by the Stiefel–Whitney classes:

$H^{*}(\operatorname {BO} (n);\mathbb {Z} _{2})=\mathbb {Z} _{2}[w_{1},\ldots ,w_{n}].$

Infinite classifying space

The canonical inclusions $\operatorname {O} (n)\hookrightarrow \operatorname {O} (n+1)$ induce canonical inclusions $\operatorname {BO} (n)\hookrightarrow \operatorname {BO} (n+1)$ on their respective classifying spaces. Their respective colimits are denoted as:

$\operatorname {O$ :=\lim _{n\rightarrow \infty }\operatorname {O} (n);}

$\operatorname {BO$ :=\lim _{n\rightarrow \infty }\operatorname {BO} (n).}

$\operatorname {BO}$ is indeed the classifying space of $\operatorname {O}$.

See also

• Classifying space for U(n)
• Classifying space for SO(n)
• Classifying space for SU(n)

Literature

• Milnor, John W.; Stasheff, James D. (1974). Characteristic Classes. Annals of Mathematics Studies. Vol. 76. Princeton University Press; University of Tokyo Press. ISBN 978-0-691-08122-9. MR 0440554.
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1990-02-21). Spin Geometry. Princeton University Press. ISBN 9780691085425.
• Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X.
• Mitchell, Stephen (August 2001). Universal principal bundles and classifying spaces (PDF).

External links

• classifying space on nLab
• BO(n) on nLab